This is a significant improvement, in particular completely removing hypothesis REF  from the original result. We might call REF  the property of being constant on divisions, where a division of a group \(G\)  is a minimal subset \(C\subset G\)  closed under conjugation (i.e. \(g\in C\)  implies \(g^h\in C\) ) and under invertible powers (i.e. if \(n\)  is coprime to the order of \(G\)  and \(g\in C\) , then \(g^n \in C\) ). Previously studied methods for counting number fields ordered by an arbitrary invariant (such as Wright's work on abelian extensions [1]}) do not require this assumption, suggesting that it is merely an artifact of the methods being used in [2]}. By using harmonic analysis to bypass this condition, we verify this intuition.
